WEBVTT 00:01.540 --> 00:06.760 In this session, we will discuss about the mathematics behind logistic regression. 00:07.330 --> 00:09.640 So first of all, let us look at a few items. 00:10.120 --> 00:12.310 So the first stop is ORMs. 00:14.330 --> 00:22.220 The odds are the chances that something will happen now, if you flip a coin, the odds are 50/50, 00:22.760 --> 00:24.360 that is you'll get hit. 00:24.980 --> 00:31.190 Now, if something strange happens, people will often see what were the odds of that, which means 00:31.400 --> 00:33.480 that I can't believe that happened. 00:33.680 --> 00:35.340 The odds were against it. 00:35.780 --> 00:37.500 So how would we find that? 00:37.670 --> 00:48.230 So, for example, the odds in favor of a ruling or two or three or six cited by one is to five or one 00:48.380 --> 00:49.460 divided by five. 00:49.760 --> 00:58.460 That is odds against will be and against an even equal number of unfavorable outcome divided by the 00:58.460 --> 00:59.960 number of favorable outcome. 01:00.230 --> 01:10.850 So here, let us say, the number of times our team has already won is one that a team has lost the 01:10.850 --> 01:12.370 game four times. 01:12.770 --> 01:19.820 So the odds in favor of the first team winning is one is to fourth with five wins. 01:19.820 --> 01:21.400 The total number of games. 01:21.650 --> 01:24.360 This is also written as one is to four. 01:25.130 --> 01:26.810 Now, what is probability? 01:27.170 --> 01:31.190 Probability is the odds of something happening or what? 01:31.190 --> 01:34.700 All the popular cases, all the possible cases. 01:34.880 --> 01:43.430 So probability in this case will be one to five where one is the chance of this team winning, divided 01:43.430 --> 01:45.830 by the total number of cases with just five. 01:46.100 --> 01:53.330 While Vilvoorde is not equal probability itself, it is the ratio of something happening over something 01:53.600 --> 01:54.700 not happening. 01:55.070 --> 02:00.900 Probability is ratio of something happening over all the possible cases. 02:00.920 --> 02:03.710 So this is the difference between odds and probability. 02:04.640 --> 02:10.790 OK, now let us see, we have odds of winning as five to three. 02:11.210 --> 02:17.150 That is five times IBM has one and three times the team has lost the game. 02:17.690 --> 02:19.490 So what will be the odds value? 02:19.490 --> 02:21.780 The value will be five divided by three. 02:21.800 --> 02:23.430 That is one point seven. 02:24.230 --> 02:26.480 Now, what will be the probability of winning? 02:26.690 --> 02:34.010 The probability of winning will be five, but just the same as the giants of the team winning, divided 02:34.010 --> 02:39.140 by the total number of values that this five plus three comes out to be eight. 02:40.420 --> 02:44.830 So the probability of winning will be five by eight here. 02:46.110 --> 02:53.880 Which is zero point six to five now, because probability of winning is equal to opposite of probability 02:53.880 --> 02:57.210 of losing, that is one minus probability of losing. 02:57.600 --> 03:00.930 So the probability of losing will be three by eight. 03:01.860 --> 03:03.760 That is zero point three seven five. 03:04.080 --> 03:09.140 We can also write as a probability of losing equal to one minus probability of winning. 03:09.840 --> 03:16.440 So which will be equal to one minus zero point sixty five, equal to zero point three seven five. 03:17.390 --> 03:25.550 Right, so probability of winning is zero point six dubi, but just five by eight probability of losing 03:25.550 --> 03:33.620 is three by eight, and probability of losing can also be written as one minus probability of winning. 03:34.920 --> 03:42.870 Now, probability of winning divided by probability of losing candidate A. probability of winning, 03:42.870 --> 03:45.120 divided by one minus probability of winning. 03:46.020 --> 03:50.640 We have just replaced probability of losing by one minus probability of winning. 03:51.630 --> 03:59.700 So this gives us five by eight, divided by three by which comes out to be five by three, which is 03:59.700 --> 04:02.760 actually what it does, the odds of winning. 04:03.740 --> 04:10.990 So you can say that probability of winning divided by probability of losing is equal to or winning, 04:11.720 --> 04:12.070 right? 04:12.320 --> 04:22.280 So if the goal, the probability of winning as fee, then we can also be divided by one minus the. 04:24.340 --> 04:32.860 You can take some moment for the slide and try to understand what I just said, that is the what is 04:32.860 --> 04:33.960 the odds of winning? 04:33.970 --> 04:35.830 What is the probability of winning? 04:36.100 --> 04:42.200 How probability of winning is related to the probability of losing and all of this mathematics. 04:42.970 --> 04:47.470 Now, the odds value, which is five, is to three. 04:47.710 --> 04:51.400 This odds value will range from zero to one. 04:52.480 --> 04:58.450 For against scenario, they range from one to infinity for favorite movies. 04:59.320 --> 05:01.840 So let's see if. 05:02.960 --> 05:09.140 Our theme is the value of is if the above value, if the odds of winning. 05:10.640 --> 05:16.050 Is a larger value rate, the odds of winning is a larger value here. 05:16.280 --> 05:24.680 So in this particular case, the favorable case will be of greater value divided by the smaller value. 05:25.500 --> 05:31.590 A favorable case will always be of greater value divided by the smaller value, then only it will be 05:31.590 --> 05:33.550 the in favor of the being right. 05:33.840 --> 05:40.950 So it will always be a greater number divided by a smaller number, which means that it will always 05:40.950 --> 05:42.030 be greater than one. 05:43.630 --> 05:47.890 So the value greater than one can range from one to infinity. 05:48.670 --> 05:56.740 Now when we are discussing both against scenario so four against scenario, it will always be a smaller 05:56.740 --> 05:58.390 dome divided by the larger. 06:01.280 --> 06:09.020 So when this would happen, then the value will be between zero to one and a smaller number divided 06:09.020 --> 06:15.020 by a larger number, will always give us a value which is ranging from zero to one. 06:15.950 --> 06:23.540 So this is what we get from or so of value, which is ranging from zero to one, is always be present 06:23.540 --> 06:27.200 for or against scenario and for favorable scenario. 06:27.200 --> 06:31.520 Odds will always range from one to infinity. 06:33.350 --> 06:33.860 Now. 06:39.670 --> 06:41.560 Now, when you see this. 06:42.440 --> 06:43.910 Four hours of winning. 06:44.840 --> 06:46.430 So although veni. 06:47.500 --> 06:50.920 Will be a favorable scenario right now. 06:51.060 --> 06:51.930 Let's go for the. 06:54.310 --> 07:03.310 Now, let us see, we have these equations which we have formulated, which is probability of Y is equal 07:03.310 --> 07:10.810 to one given the excess true is B they we just said we will say probability of winning. 07:13.500 --> 07:24.000 What is being that this is probability of Y is equal to one given X is equal to B, then law will be 07:24.900 --> 07:31.770 divided by one minus V will be better not let's be double next one plus X. 07:33.300 --> 07:37.200 So here what we are doing is we have this equation. 07:40.150 --> 07:46.960 Where we were seeing that the probability is equal to be done or plus with our next one, plus we do 07:46.960 --> 07:49.970 X to be the extra and so on. 07:50.350 --> 07:56.950 So we are seeing that this is the probability value now because this probability value was ranging from 07:56.950 --> 07:57.780 zero to one. 07:58.060 --> 08:02.660 And this equation has a range from minus infinity to infinity. 08:02.950 --> 08:12.430 So what we can do is we can convert this equation into the same range value by taking a log of. 08:13.550 --> 08:19.060 So if we will meet the goal, all of it, then the range mismatch will be resolved. 08:20.100 --> 08:28.020 OK, so what we do here, we will take a look and now for taking a look at the situation and on solving 08:28.020 --> 08:31.090 the equation for VA, we get the equation. 08:31.150 --> 08:37.910 The VA is equal to one upon one plus E to the power minus Vetinari plus vedova an excellent as that 08:38.280 --> 08:38.880 went on. 08:39.390 --> 08:44.700 And this is actually the logic function which we have discussed about. 08:45.210 --> 08:51.810 This is the logic function we have formed for this equation, which will convert this linear line into 08:51.810 --> 08:55.530 this goal line, which will be able to map to the. 08:56.710 --> 08:58.210 Two classes, which we have. 09:00.430 --> 09:04.600 So why do we do this with this equation now? 09:04.810 --> 09:12.070 The objective here is not to correctly estimate the value of log the by one minus B, so this is not 09:12.070 --> 09:14.290 the objective V one. 09:14.290 --> 09:22.540 Do either maximize or minimize the value of B such that we get either a low probability or higher probability 09:22.540 --> 09:27.100 that it's either somebody would just close to zero or somebody would just close to one. 09:27.910 --> 09:36.130 So here we will be having the probability value as some value as zero point one zero point three zero 09:36.130 --> 09:37.370 point five zero point seven. 09:37.390 --> 09:42.010 So these would be different probability values, which will be ranging from zero to one. 09:42.550 --> 09:44.930 Now we have these probability values. 09:45.160 --> 09:53.010 Now we want to change the values such that when we take this log of this, the this function will actually 09:53.040 --> 09:56.350 tell the values closer to zero and one. 09:56.740 --> 10:04.950 So the values will be if the values closer to zero, then it is more prominent that the answer is zero. 10:05.080 --> 10:06.610 That is because of zero. 10:06.610 --> 10:13.420 Plus, if the value of the probability is closer to one, that means that the answer is one. 10:13.420 --> 10:17.350 That is the probabilities of the first class. 10:19.110 --> 10:27.720 Now we have this likely war to function, but just create it so that we can formulate the same thing. 10:29.370 --> 10:38.220 So here, the objective is not to correctly estimate the value of log be divided by one minus five. 10:38.740 --> 10:48.210 OK, so we want that barometer's to be values which reside in such a score or a probability which enables 10:48.210 --> 10:50.490 us to have a good cut of value. 10:50.940 --> 10:59.610 So if the values will be closer to zero or closer to one, then we can have a lot of value which will 10:59.610 --> 11:03.090 be more prominent, which will be more helpful. 11:03.240 --> 11:09.740 If the total value is not created properly, then some points might be misclassified. 11:09.930 --> 11:12.330 And that is something which we don't want to happen. 11:14.100 --> 11:20.140 So meaning this school should be high for one class and low for another class. 11:20.160 --> 11:26.010 We want to have a good distance between both of these school values. 11:26.920 --> 11:29.500 Now, let us consider this particular equation. 11:32.040 --> 11:41.520 So here, when Y is equal to zero, when Y will be zero at that point of time, the value of likelihood 11:41.520 --> 11:43.590 this function is called likelihood function. 11:43.800 --> 11:48.600 So when Y will be zero, the equation will go on and do one minus B. 11:49.910 --> 11:58.040 Because this value will be zettl, so this dome will turn into one, this value will be one minus one. 11:58.050 --> 11:59.320 So this will turn into. 12:01.210 --> 12:07.590 This this will be one minus zero, so it will turn into one, so we will get one minus B as the equation 12:08.080 --> 12:12.130 now when the value of Vye will be one. 12:12.280 --> 12:16.090 So what will happen to the other one will be probability. 12:16.390 --> 12:19.190 Now, this will be one minus one zero. 12:19.510 --> 12:22.690 So this will give up one here. 12:23.470 --> 12:30.340 So that is why the entire tone of likelihood function will go on and do the. 12:35.950 --> 12:44.650 So in other words, we want to maximize this likelihood function, we want to achieve this value because 12:44.650 --> 12:51.760 if we will maximize the likelihood function, then the value of likelihood will go into either B or 12:51.760 --> 12:52.540 my one mind. 12:52.540 --> 12:57.250 The speed, which is not a good hit that is then is equal to zero. 12:57.250 --> 13:02.410 The likelihood equals probability of being one then by is equal to zero. 13:02.410 --> 13:07.220 The likelihood of probability will be of Y will be zero. 13:07.600 --> 13:11.160 So we want that probabilities to match with the real outcome. 13:11.440 --> 13:14.990 Hence we want to maximize the value of L. 13:15.250 --> 13:17.410 So that is the value. 13:17.410 --> 13:21.700 When Y is equal to one, the likelihood will be probability. 13:21.910 --> 13:29.920 That is, it will go on and do one value of maximum value and then the negative form will diminish. 13:30.130 --> 13:31.510 The storm will diminish. 13:31.750 --> 13:39.700 And when the value of Y is equal to zero, then this entire goal will be reduced to almost zero. 13:40.300 --> 13:42.760 So this is what the likelihood will do to us. 13:43.030 --> 13:47.650 Now, this function which we have created is for only one observation. 13:48.760 --> 13:54.560 So we will have to convert it into a combination of all the observation. 13:55.570 --> 14:05.110 So for this, we can do this by basically multiplying all of these storms together, because in terms 14:05.110 --> 14:09.520 of probability, then we want to combine different probabilities. 14:09.520 --> 14:11.860 We will be multiplying these probabilities. 14:12.190 --> 14:19.300 I in earlier scenario, when we were combining the different medications for different rules of beta, 14:19.600 --> 14:20.860 we do the same mission. 14:21.130 --> 14:23.710 But in in this case, we have probabilities. 14:23.710 --> 14:27.070 So we will be taking a product of the probabilities. 14:28.850 --> 14:32.330 So we will have to maximize this collective form. 14:32.510 --> 14:34.880 We will have to maximize this entire thing. 14:35.660 --> 14:40.010 So for maximizing this and the entire thing, we will be. 14:41.380 --> 14:47.860 Then voting this into a cost function and we will be taking the logo of this entire thing and this becomes 14:47.860 --> 14:50.410 the cost function of this entire. 14:51.290 --> 14:57.470 Of likelihood function for logistic regression, this is the cost function of the logistic regression. 14:58.400 --> 15:04.340 So next is different performance metrics, which we will be launching later. 15:04.960 --> 15:12.200 But for now, what do you need to understand is that even if you don't understand the more details of 15:12.200 --> 15:19.790 the entire concept here, which we have just discussed, you just need to remember that our main purpose 15:19.820 --> 15:24.500 is to throw in the values we want to get the probability value. 15:25.620 --> 15:29.700 Which is closer to zero, and one that is because. 15:30.970 --> 15:33.220 When we have this kind of plot. 15:34.160 --> 15:42.890 When we have this kind of plot, our main target is to set up a value which is a cartel value. 15:44.740 --> 15:50.470 Now, the current value will supposedly be in the middle of this line. 15:51.310 --> 15:57.940 So when the value is in the middle of this line, what we want to achieve is if the probability value 15:57.940 --> 16:04.570 will be very low and very high, then what will happen is wherever we will put the cost of value, these 16:04.570 --> 16:07.060 points will not get misclassified. 16:08.530 --> 16:17.260 The line will be more towards if the cutoff, if the distance is more, if the distance between these 16:17.260 --> 16:20.350 is more, then the line would do something like. 16:22.330 --> 16:22.870 This. 16:24.530 --> 16:29.750 So this will be able to make a better decision, give a better decision boundary. 16:31.280 --> 16:38.410 So that is what we want to achieve instead of having the the values of the probabilities very close. 16:38.630 --> 16:40.980 We want to have the value of probabilities. 16:41.000 --> 16:47.210 Finally, when the values of probabilities will be far away, then the value can range from anywhere 16:47.210 --> 16:48.590 to anywhere like this. 16:48.860 --> 16:55.300 And we will still be able to classify the data properly because now the data is far from each other, 16:55.520 --> 16:57.950 the probability values are far from each other. 16:57.980 --> 17:01.200 So any cutoff will still make that magnetism. 17:02.510 --> 17:08.720 But when those values will be closer like this, when the values will be closer like this. 17:09.680 --> 17:14.750 In that case, if the values are closer like this, somewhat like this. 17:16.210 --> 17:22.200 Then the cutoff will have to be a very strict one, then it will be very important to find the perfect 17:22.210 --> 17:23.030 cutoff for you. 17:23.620 --> 17:31.030 But in this scenario, because the values are already far away, we can have any of value between this 17:31.030 --> 17:35.170 to this end of value will give a good result. 17:35.890 --> 17:37.860 So that is what we want to obtain. 17:38.080 --> 17:45.310 And the only formula that you want to remember from this entire thing is the formula for the logic function 17:45.670 --> 17:47.860 that is being equal to. 17:48.980 --> 17:55.760 One upon one plus E to the power be done on Glasby, Delmonico's one, Buzby that way too and so on, 17:56.120 --> 18:02.780 when this be done will be the one that will be the professions which you will be generating an X one 18:02.780 --> 18:09.920 extra extra hour, different values of the features which we have, the feature values, the attributes 18:09.920 --> 18:10.660 which we had. 18:10.910 --> 18:13.550 So it is similar to the linear regression. 18:13.550 --> 18:19.430 But the only thing is we have applied this logistic function instead of the normal linear regression 18:19.700 --> 18:23.120 so that we can get this kind of transformation here. 18:23.450 --> 18:27.950 This is the only thing which is important for you to understand here. 18:29.380 --> 18:36.400 Now, in the next session, we will learn about different types of metrics which are available, and 18:36.400 --> 18:43.120 then we will go ahead with the implementation of this so that you will get a better picture of what 18:43.180 --> 18:44.430 you are trying to do here.